The oldest civilizations to produce beer were the Sumerians and Egyptians. Their beer was fermented in huge stone jars that held hundreds of liters before being transferred to smaller containers. The Sumerians, at least, often drank beer through reed straws to avoid sucking up sediment left in the container. I have a picture of a Hittite beer jar that has a trough-like spout with relatively large filter holes for pouring, so there were attempts at removing the solids but they weren't great. There are still many traditional African beers made with millet, sorghum, and other grains that aren't far off from these ancient brews.

By the time ale/beer are the main drink in medieval Europe ("beer" in the sense of an ale flavored with hops reached England around 1400, before hops became the main bittering/flavoring/preservation agent in the 16th century), it was transferred to wooden barrels that would be tapped into a tankard for serving. These beers would have only the natural carbonation provided by the yeast producing carbon dioxide in addition to alcohol and refrigeration was nonexistent, so it would have been relatively flat and served at room temperature or cellar temperature. "Real ale" in the UK is preferably served this way, albeit in metal kegs rather than wooden barrels today.

Prohibition in the United States effectively killed the cooperage industry, as alcohol was one of the few products still being packed into traditional wooden barrels at the time. When Prohibition ended in 1933, there were insufficient barrels to hold the beer that was now being brewed again. Aluminum kegs were employed as an industrial solution, as it was cheap and light, but it was quickly replaced by stainless steel due to its need for epoxy linings to avoid corrosion from alcohol and caustic soda cleaning solutions.

The 20th century saw the advent of modern stainless steel kegs and cleaning/filtering systems, which allowed for even unpasteurized beer to be effectively transferred to kegs for draft/draught (same pronunciation, different level of Britishness) systems. These systems also easily allow for pressurization with CO2 and/or nitrogen to provide carbonation and a foamy head.

As far as beer outside the tap goes, it was initially sold right from the keg into whatever container you had. Galvanized pails with lids were common in the 19th and early 20th century; they earned the nickname "growler" from the sound made by CO2 escaping from imperfectly sealed lids.

Bottled beer was actually pioneered by Miller High Life, which sold "the Champagne of Beers" in a champagne-style glass bottle on shelves to highlight its status as a premium product. Cans became popular after Prohibition, initially either with tapered tops that fit a standard bottle cap or flat tops that required a "church key" can opener to puncture a hole in the top. These would evolve into pull tabs in the 1960s, which peeled completely off the can. For child safety reasons, in the 1970s these would be replaced by the modern stay-tabs.

If you want way more details than you ever thought you'd need, I recommend buying The Oxford Companion to Beer as a handy reference book.